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With the help of support functions of fuzzy sets, we are able to embed some interesting families of fuzzy sets into Banach spaces. It is better to make the embedded Banach spaces as small as possible, which can be achieved by considering the weak* topology. In this case, the weak* continuity of support functions should be studied. The concept of a maximal embedding behavior is also presented in this paper, which shows the maximal subset satisfying the completeness.

Abstract We obtain sharp inequalities of Hardy type for functions in the Sobolev space W 1 , p $W^{1,p}$ on the unit sphere S n − 1 $\mathbb{S}^{n-1}$ in R n $\mathbb{R}^{n}$ . We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities. We apply our results to obtain lower bounds for the the first eigenvalue of the p-Laplacian on the sphere.